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arXiv:1212.0105

The Vector representation of the Standard Quantum Process Tomography

Xiaohua, Wu

Published Dec 1, 2012• 0 citations
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Abstract

The characterization of the evolution of a quantum system is one of the main tasks to accomplish to achieve quantum information processing. The standard quantum process tomography (SQPT) has the unique property that it can be applied without introducing any additional quantum resources. In present work, we shall focus on the following two topics about the SQPT. At first, in the SQPT protocol for a $d$-dimensional system, one should encounter a problem in solving of a set of $d^4$ linear equations in order to get the matrix containing the complete information about the unknown quantum channel. Until now, the general form of the solution is unknown. And a long existed conviction is that the solutions are not unique. Here, we shall develop a self-consistent scheme, in which bounded linear operators are presented by vectors, to construct the set of linear equations. With the famous Cramer's rule for the set of linear equations, we are able to give the general form of the solution and prove that it is unique. In second, the central idea of the SQPT is to prepare a set of linearly independent inputs and measuring the outputs via the quantum state tomography (QST). Letting the inputs and the measurements be prepared by two sets of the rank-one positive-operator-valued measures [POVMs], where each POVM is supposed to be linearly independent and informationally complete (IC), we observe that SQPT now is equivalent to deciding a unknown state with a set of product IC-POVM in the $d^2$-dimensional Hilbert space. Following the general linear state tomography theory, we show that the product symmetric IC-POVM should minimize the mean-square Hilbert-Schmidt distance between the estimator and the true states. So, an optimal SQPT can be realized by preparing both the inputs and the measurements as the symmetric IC-POVM.

DOI
10.48550/arxiv.1212.0105

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Quantum State Tomography (QST), Quantum Process Tomography (QPT), and Quantum Network Tomography (QNT) are related parameter-estimation problems that aim to reconstruct different physical quantities. QST estimates an unknown quantum state, represented by its density matrix, from the measurement outcomes. QPT characterises an unknown quantum channel using known input states and measurements of the corresponding outputs. QNT, in contrast, aims to infer parameters associated with individual links from end-to-end probe measurements collected at accessible monitor nodes. A key distinction among the three tomography problems lies in the conditions required to achieve identifiability, the ability to determine unknown parameters uniquely from the available measurement statistics. In QST and QPT, the experimenter can choose an Informationally Complete (IC) measurement set. QNT limits the reachable measurements to what topology and monitor placement allow, so the admissible probe paths fix the information available about the link parameters. This work studies all three tomography problems through a common Fisher Information Matrix (FIM). We factorise QNT FIM and show that its rank equals the rank of the path-link incidence matrix at every interior parameter value, so local and global identifiability coincide. We then show that QST and QPT attain full rank under IC settings, QNT loses rank when the probe paths leave link parameters indistinguishable, and increasing the number of copies scales the FIM eigenvalues while leaving its rank fixed.
Recently, Bendersky \emph{et al.} developed a method to complete the task of characterizing an arbitrary χχχ matrix element in a scalable way, Phys. Rev. Lett. Vol. \textbf{100}, 190403(2008), where an auxiliary system was needed. In present work, we shall show that the same task can also be completed within the scheme of standard quantum process tomography (SQPT) where there is no requirement for ancilla. Our method depends on two observations: With the elaborately chosen operators basis, the SQPT may have an economical form where a single run of experiment, in which we measure the expectation value of a chosen operator in the outport of the quantum channel with a known input, is sufficient to characterize a selected χχχ matrix element; With the progress recently achieved in quantum entanglement detection, we also find that the number of the experimental settings to realize the experiment for the selected χχχ matrix element does not exceed 2N for the N-qubits system. For practice, our scheme can be applied for the cases where the controlled two-body interaction is neither available nor desirable.
Quantum process tomography (QPT) methods aim at identifying a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. The present paper focuses on the estimation of a unitary process. This class is of particular interest because quantum mechanics postulates that the evolution of any closed quantum system is described by a unitary transformation. Unitary processes have significantly fewer parameters than general quantum processes (22nqb vs. 24nqb−22nqb real independent parameters for nqb qubits). By assuming that the process is unitary we develop two methods that scale better with the size of the system. In the present paper, we stay as close as possible to the standard setup of QPT: the operator has to prepare copies of different input states. The properties those states have to satisfy in order for our method to achieve QPT are very mild. Therefore, we choose to operate with copies of 2nqb initially unknown pure input states. In order to perform QPT without knowing the input states, we perform measurements on half the copies of each state, and let the other half be transformed by the system before measuring them (each copy is only measured once). This setup has the advantage of removing the issue of systematic (i.e., same on all the copies of a state) errors entirely because it does not require the process input to take predefined values. We develop a straightforward analytical solution that first estimates the states from the averaged measurements and then finds the unitary matrix (representing the process) coherent with those estimates by using our analytical solution to an extended version of Wahba’s problem. This estimate may then be used as an initial point for a fine tuning algorithm that maximizes the likelihood of the measurements. Simulation results show the effectiveness of the proposed methods.
The temporal evolution of a quantum system can be characterized by quantum process tomography, a complex task that consumes a number of physical resources scaling exponentially with the number of subsystems. An alternative approach to the full reconstruction of a quantum channel is the selective and efficient quantum process tomography, a method that allows estimating, individually and up to the required accuracy, each element of the matrix that describes the process, using only a polynomial amount of resources. The implementation of this protocol is closely related to the possibility of building a complete set of mutually unbiased bases (MUBs), whose existence is known only when the dimension of the Hilbert space is the power of a prime number. However, an extension of the method that uses tensor products of maximal sets of MUBs has been recently introduced. Here we explicitly describe how to implement the algorithm for a selective and efficient estimation of a quantum process in a nonprime power dimension and conduct, an experimental verification of the method in a Hilbert space of dimension d=6d=6d=6. This is the smallest space for which a complete set of MUBs is not known to exist, but it can be decomposed as a tensor product of two Hilbert spaces of dimensions D1=2{D}_{1}=2D1​=2 and D2=3{D}_{2}=3D2​=3, in which a complete set of MUBs is already known. The six-dimensional states were codified in the discretized transverse linear momentum of photons. The state preparation and detection stages are dynamically programed with the use of only-phase spatial light modulators, in a versatile experimental setup that allows one to implement the algorithm in any finite dimension.
Quantum process tomography (QPT) methods aim at identifying, i.e., estimating, a given quantum process. QPT is a major quantum information processing tool, since it especially allows one to characterize the actual behavior of quantum gates, which are the building blocks of quantum computers. However, usual QPT procedures are complicated, since they set several constraints on the quantum states used as inputs of the process to be characterized. In this paper, we extend QPT so as to avoid two such constraints. On the one hand, usual QPT methods require one to know, hence to very precisely control (i.e., prepare), the specific quantum states used as inputs of the considered quantum process, which is cumbersome. We therefore propose a blind, or unsupervised, extension of QPT (i.e., BQPT), which means that this approach uses input quantum states whose values are unknown and arbitrary, except that they are requested to meet some general known properties (and this approach exploits the output states of the considered quantum process). On the other hand, usual QPT methods require one to be able to prepare many copies of the same (known) input state, which is constraining. In contrast, we propose ``single-preparation BQPT methods'' (SBQPT) , i.e., methods which can operate with only one instance of each considered input state. These two concepts are here illustrated with practical (S)BQPT methods which are numerically validated, in the case when (i) random pure states are used as inputs and their required properties are especially related to the statistical independence of the random variables that define them and (ii) the considered quantum process is based on cylindrical-symmetry Heisenberg spin coupling. As a benchmark, we moreover introduce nonblind QPT methods dedicated to the considered Heisenberg process, we analyze their theoretical behavior (this requires the tools developed in this paper for random input states), and we numerically test their sensitivity to systematic and nonsystematic errors, which are most likely to occur in practice. This shows that, even for very low preparation errors (especially systematic ones), these nonblind QPT methods yield much lower performance than our SBQPT methods. Our blind and single-preparation QPT concepts may be extended, e.g., to a much wider class of processes and to SBQPT methods based on other quantum state properties, as outlined in this paper.
Characterization of quantum processes is a preliminary step necessary in the development of quantum technology. The conventional method uses standard quantum process tomography, which requires d2{d}^{2}d2 input states and d4{d}^{4}d4 quantum measurements for a ddd-dimensional Hilbert space. These experimental requirements are compounded by the complexity of processing the collected data, which can take several orders of magnitude longer than the experiment itself. In this paper we propose an alternative self-guided algorithm for quantum process tomography, tuned for the task of finding an unknown unitary process. Our algorithm is a fully automated and adaptive process characterization technique. The advantages of our algorithm are the following: it has an inherent robustness to both statistical and technical noise; it requires less space and time since there is no postprocessing of the data; it requires only a single input state and measurement; and it provides on-the-fly diagnostic information while the experiment is running. Numerical results show our algorithm achieves the same 1/n1/n1/n scaling as standard quantum process tomography when nnn uses of the unknown process are used. We also present experimental results wherein the algorithm and its advantages are realized for the task of finding an element of SU(2).